Excel in your current position
Exceed the expectations set for you where you are. If you are looking to be ambitious in the next position, you must be as ambitious as possible with your current one. This is always possible unless your employer has set unattainable expectations for you. It may be difficult to see how it is possible, but if you benchmark out what you do it shouldn't be difficult to find a couple of things that would make this possible.
Do things outside your current position that relate to the new one
If there are any tasks involved in the position you would like to pursue that are even remotely relevant to your current position, begin doing those things as much as possible. Make yourself available and obvious.
Talk to leadership
Your supervisor/manager should not be there to hold you back or to keep a stagnant team. If your boss is at all effective, he/she will want to do whatever it takes to help you achieve your goals within the company because making you better makes the company better. Talk with your supervisor. Tell him/her that you believe you can be so much more. Ask what you can do to prepare for the new position. Ask if your boss knows individuals who can help you prepare or guide you.
Talk to someone who actually holds the position (find a mentor)
Someone must be doing the job you're looking to get. Talk to someone and ask them what they did to get the qualifications to do the job. Ask them for pointers on how to accelerate those qualifications, pitfalls to avoid, even people to schmooze if it comes to that. Sometimes getting a start in a position you're not qualified for comes down to who you know that is willing to take a chance on you.
Seek help from friends
If you have any friends or acquaintances who can help you in this endeavor use those connections. Even if it's just having a friend whose sole assistance is helping to keep you focused while you attend classes to receive training. Don't be afraid or too proud to seek out this help. When the time comes, don't be stingy with helping them either.
If this is what you want, then you have to make it yours. When reading your statement:
However, due to the equally awkward hours, there is very little time to train myself, and learn new skills., it rings as an excuse to me. It may be that there is little time and juggling your schedule is difficult. But it's not impossible, so make it happen. It may mean that other parts of your life lose out while you accomplish this, but if this is what you want then you have to put the excuses away and get it done. Note: This part is not meant to be accusatory to OP. I don't know the specific challenges standing in the way. This section is meant as general advice to anyone in a similar position who seems to find one obstacle after another.